A waveplate alters polarized light by adding a predetermined phase shift between two orthogonal polarization components of the polarized light field. Conventionally, the added phase shift is referred to as the waveplate retardance and is measured in fractions of a wavelength for normal incident light. For example, a waveplate that adds a phase shift of π between the orthogonal polarizations is referred to as a half-wave plate, whereas a waveplate that adds a phase shift of π/2 is referred to as a quarter-wave plate.
Traditionally waveplates have been fabricated using uniaxial birefringent materials. A uniaxial birefringent material has two principal refractive indices, namely an ordinary refractive index no and extraordinary refractive index ne, where the birefringence is defined as Δn=ne−no. In a waveplate, the axis having the low refractive index is termed the fast axis, whereas the axis having the higher refractive index is termed the slow axis and is at a right angle to the fast axis. For example, for a positive uniaxial birefringent material, where ne>no, the extraordinary axis is the slow axis whereas the ordinary axis is the fast axis. The extraordinary axis is also the optic axis, which is understood to be the direction in which linearly polarized light propagates through the medium with the same speed, regardless of its state of polarization. In order to provide the required phase shift, the optic axis of uniaxial birefringent materials used in the fabrication of conventional waveplates is typically oriented at a non-normal angle to the plane of the waveplate. For example, waveplates are often fabricated from a uniaxial birefringent material having its optic axis parallel to the plane of the waveplate (i.e., termed an A-plate) or at an oblique angle to plane of the substrate (i.e., termed an O-plate). Alternatively, a waveplate may be fabricated from a uniaxial birefringent material having its optic axis normal to the plane of the waveplate (i.e., termed a C-plate) such that its retardance increases with angle of incidence (AOI).
One important application of waveplates is to alter the polarization state of polarized light travelling through it. For example, half-wave plates can be used to rotate the orientation of linearly polarized, whereas quarter-wave plates can be used to convert linearly polarized light to circularly polarized. With regard to the former, the rotation angle is 2θ when the incident light's polarization direction is oriented at θ to the optic axis in the plane of the waveplate. With regard to the latter, the incident light's polarization direction is typically oriented at 45 degrees to the optic axis in the plane of the waveplate. In each case, it is preferred that the optic axis of the waveplate be spatially uniform (i.e., non-varying across a surface of the waveplate such that the entire surface of the waveplate has the same optic axis orientation) in order to provide uniformly-polarized beams (i.e., beams of polarized optical radiation in which the polarization does not vary across the beams' cross-section), assuming the incident polarization state is also uniform across it's cross section.
More recently, it has been recognized that inducing spatial polarization variations across a uniformly polarized beam is an invaluable wavefront shaping tool. In fact, if such a beam with space-variant polarization is analyzed using a linear polarizer, the net effect is the addition of a spatially-variant phase shift, known as the Pancharatnam-Berry phase, across the beam's cross-section. Some examples of light beams having a spatially-varying linear polarization are radially-polarized and azimuthally-polarized (i.e., tangentially-polarized) light beams, in which the local axis of polarization is either radial, that is, parallel to a line connecting a local point to a center point, or tangential, that is, perpendicular to that line. The polarization patterns of these beams are illustrated in FIG. 1, wherein arrows schematically show local orientations of the beam polarization.
Whether the beam is radially or azimuthally polarized, its polarization direction depends on an azimuth angle of a particular spatial location and does not depend on the radial distance from the center point. These types of polarized beams are sometimes referred to as cylindrical vector beams or polarization vortex beams. The term “polarization vortex” is related to the term “optical vortex”. An optical vortex is a point in a cross-section of a beam that exhibits a phase anomaly so that the electrical field of the beam radiation evolves through a multiple of π in any closed path traced around that point. Similarly, a polarization vortex is a linearly polarized state in which the direction of polarization evolves through a multiple of π about the beam axis. Such a beam, when focused, adopts a zero intensity at the center point (e.g., along the beam's axis if the vortex is centered within the beam). Polarization vortex beams have a number of unique properties that can be advantageously used in a variety of practical applications such as particle trapping (optical tweezers), microscope resolution enhancement, and photolithography.
One method of obtaining a polarization vortex beam is to pass a uniformly polarized optical beam through an optical vortex retarder. An optical vortex retarder, which is also referred to herein simply as a vortex retarder, refers to a class of waveplates that has a spatially varying fast axis that rotates around a point. More specifically, an azimuthal angle of the fast axis rotates about a point. If the optical retarder is an achromatic retarder (e.g., a multi-layer design wherein two or more retarders are stacked or laminated in order to make the optical retarder achromatic) then the spatially varying fast axis is the effective fast axis (i.e., the orientation that would appear to the be fast axis if multi-layer retarder were assumed to be a single layer of birefringent material). The term “azimuthal angle” refers to the azimuthal orientation of the axis projected in the plane of the optical retarder, measured relative to some arbitrary reference point. Note that while the azimuthal angle of the fast axis of a vortex retarder rotates about a point, the polar angle of the fast axis is typically constant across a surface of the retarder (i.e., vortex retarders typically have a spatially uniform retardance). The term “polar angle” refers to the out-of-plate tilt of the fast axis.
In general, the spatially varying fast axis azimuth of a vortex retarder will vary with azimuthal location on the vortex retarder in a predetermined relationship. For example, referring to FIG. 2, the spatially varying fast axis azimuth θ typically varies with azimuthal location φ according to:θ(φ)=αφ+θ(0)  (1)
where α is a constant equal to the rate of change in fast axis azimuth with respect to azimuthal location. Note that both the fast axis azimuth θ and the azimuthal location φ are measured relative to a predetermined reference point (e.g., shown as the x-axis). The fast axis azimuth at this reference point is θ(0). When θ(0)=90 degrees as illustrated in FIG. 3A the fast axis is said to be tangentially-aligned. When θ(0)=0 degrees as illustrated in FIG. 3B the fast axis is said to be radially-aligned.
Referring again to Equation (1), the spatially varying fast axis θ will be only continuous at all φ if α=m/2, where m is an integer referred to as the mode of the vortex retarder. In fact, vortex retarders are often characterized according to their mode (e.g., m=2α). For example, the vortex retarders illustrated in FIGS. 3A and 3B are m=2 vortex retarders. Notably, m=2 vortex retarders (e.g., wherein α=1) correspond to the special case wherein a 1 degree counter clockwise rotation in azimuthal location corresponds to a 1 degree increase in fast axis azimuth orientation. In contrast, in a m=−2 vortex retarder (e.g., wherein α=−1) a 1 degree counter clockwise rotation in azimuthal location corresponds to a 1 degree decrease in fast axis azimuth orientation. FIG. 4 shows examples of vortex retarders having modes equal to 1, −1, 2, and −2.
Vortex retarders have been fabricated using a series of birefringent crystals, stress induced birefringence, nanostructures, liquid crystals (LC), and liquid crystal polymers (LCP). The use of LC and LCP for fabricating vortex retarders is advantageous because the resulting vortex retarders are useful in the visible wavelength range and have a continuously varying fast axis. Moreover, both LC and LCP materials can be aligned using a linear photopolymerizable polymer (LPP) layer, which is photosensitive to linearly polarized ultraviolet (LPUV) light. More specifically, the LPP layer is selectively polymerized in the direction parallel to LPUV light. Accordingly, a vortex retarder can be fabricated by rotating at least one of the substrate supporting the LPP layer and an orientation of the LPUV light. For example, the fabrication of vortex retarders using LCP has been described in S. C. McEldowney, D. M. Shemo, R. A. Chipman, and P. K. Smith, “Creating vortex retarders using photoaligned liquid crystal polymers,” Opt. Lett. Vol. 33, 134-136 (2008) and Scott C. McEldowneyl, David M. Shemo, and Russell A. Chipman “Vortex retarders produced from photo-aligned liquid crystal polymers”, Vol. 16, 7295-7308, 2008, both of which are incorporated herein by reference.
While rotating the substrate and/or the orientation of the LPUV light while irradiating the LPP layer has been shown to provide improved vortex retarders, the method is limited to making single vortex retarders.
In J. N. Eakin and G. P. Crawford, “Single step surface alignment patterning in liquid crystals using polarization holography exposure”, SID 06, p 875, a holographic exposure technique is used to create a plurality of relatively small vortex retarders, each of which has a spatially varying fast axis that rotates about a different point. More specifically, the plurality of relatively small vortex retarders, which is configured as a two-dimensional array, is created by patterning a LPP layer using the interference pattern generated by the holographic exposure. While the two-dimensional patterning of the LPP layer is conveniently performed with a single step, it is, unfortunately, a relatively complex procedure relying on the interference of four non-coplanar coherent laser beams. In addition, since the interference pattern is used to provide the two-dimensional patterning it is difficult to control and optimize the process, and in particular, the size of the array, which is limited by the laser spot size and optics. Moreover, using the interference pattern to provide the two-dimensional patterning introduces intensity modulations which may negatively affect the spatially uniform out-of-plane tilt of the spatially varying fast axis.